Before proceeding, in order to conceptually expose our results, we discuss the fibre bundle structure and various possible situations that arise; see the guiding Figure REF . Since the factored-out subgroup \(R\)  is contained in the parabolic \(Q\) , the space \(X\)  has a natural \(G\) -equivariant projection \(\pi \)  onto the flag variety \(G/Q\) . The fibres of \(\pi \)  are copies of the quotient \(Q/R\) , and by construction, we have \(Q/R \simeq (Q/R_0)/(R/R_0)=S/\Lambda \) . Since this projection is \(G\) , and hence \(\Gamma _\mu \) -equivariant, any \(\mu \) -stationary measure \(\nu \)  on \(X\)  projects down to a \(\mu \) -stationary measure \(\overline{\nu }:=\pi _\ast \nu \)  on \(G/Q\) . It follows that a first rough classification of stationary measures is provided by the classification in the base \(G/Q\) . Thanks to the results of Guivarc'h–Raugi [1]} and Benoist–Quint [2]} (see §REF ), there are two types of projections giving rise to Case 1 and Case 2, respectively, Dirac measures and Furstenberg measures on the base. In Case 1, the works of Benoist–Quint [3]}, [4]} and Eskin–Lindenstrauss [5]}, [6]} directly apply and hence we will not comment on it further here (see §REF ).
